Extraction of regular branches

Let $\Omega$ be the plane with the segment $\left\{ -1 \leq x \leq 1, y = 0\right\}$ deleted. For which of the multi-valued functions

Let $G = D \setminus (-1,0]$, where $D = \left\{ z:\left|z\right|<1\right\}$.

Show that the function $w = \ln (z+\sqrt{z^{2}-1})$ is single-valued in the plane with cuts along the real axis for $-\infty < x < -1$ and $1 < x < \infty$. Find the values of $w$ at $z = +i$ and $z = -i$, given that on the upper edge of the right-hand cut $w$ is positive.

Prove that there exists a unique function $h(z)$, continuous on the entire complex plane with a cut along the positive part of the real axis, satisfying the conditions:

Prove that this function is regular in its domain of definition.

Let the function $f: G \rightarrow \mathbb {C}$ be regular in a domain $G$ with $f(z) \neq 0 \quad \forall z \in G$. Let the function $h: G \rightarrow \mathbb {C}$ be a continuous branch of the multivalued function $\operatorname {Ln} f(z)$. Prove that $h$ is a regular function.

Does the multivalued function $\operatorname {Ln}\left(z^{2}+z\right)$ admit regular branches in the domain $\mathbb {C} \backslash [-1,1]$?

Does the multivalued function $\left\{ \sqrt{z^{2}-z}\right\}$ admit regular branches in the domain $\mathbb {C} \backslash [-1,1]$?

Does the multivalued function $\left\{ \sqrt{z-1}\right\}$ admit regular branches in the domain $G = \left\{ z:\left|z\right| > 2\right\}$?

Does the multivalued function $\operatorname {Ln} \frac{z^{2}-1}{z^{2}+1}$ admit regular branches in the domain $G = \left\{ z:\left|z\right| > 2\right\}$?

Does the multivalued function $\left\{ \sqrt[3]{1+z^{2}}\right\}$ admit regular branches in the domain $G = \mathbb {C} \backslash [-i, i]$?

Does the multivalued function $\left\{ z^{z}\right\}$ admit regular branches in the domain $G = \mathbb {C} \backslash [0,+\infty )$?

Prove that the following multivalued functions admit the extraction of regular branches in the domains $G$ indicated in parentheses:

Let the function $f(z)$ be regular in a domain $G$ and satisfy there the condition $\mathrm{Re}\left(f(z)\right) > 0$. Prove that the following multivalued functions admit the extraction of regular branches in the domain $G$:

Let the function $f(z)$ be regular in a domain $G$ and not take there values lying on the ray $[a,+\infty ]$, where $a \in \mathbb {R}$. Prove that the multivalued function $\left\{ \sqrt{a-f(z)}\right\}$ admits the extraction of regular branches in the domain $G$, and that there exists a unique branch $\varphi (z)$ satisfying the condition $\mathrm{Re}\left(\varphi \right)(z) > 0 \quad (z \in G)$.

Let the function $f(z)$ be regular in a domain $G$ and not take values lying on a curve $\gamma$, going from the point $z = 0$ to the point $z = \infty$ while remaining in the left half-plane. Prove that the multivalued function $\operatorname {Ln} f(z)$ admits the extraction of regular branches in the domain $G$, and that there exists a branch $\varphi (z)$ for which the inequality holds

Prove that the following multivalued functions admit the extraction of regular branches in the domains $G$ indicated in parentheses:

Prove that the multivalued function

admits the extraction of regular branches in the entire complex plane with cuts along the non-intersecting straight-line segments $\left[a_{k}, b_{k}\right]$, $k = 1,2, \ldots , n$.

Prove that the following multivalued functions admit the extraction of regular branches in the indicated domains:

Prove that the multivalued function $\left\{ \sqrt[3]{1-z^{2}}\right\}$ does not admit the extraction of a regular branch in the domain $G = \left\{ z: 1 < \left|z\right| < \infty \right\}$.

Determine whether the multivalued functions given below admit the extraction of regular branches in the domains $G$ indicated in parentheses:

Determine, for what relationship between the numbers $\alpha_{1}$ and $\alpha_{2}$, the multivalued function

admits the extraction of a regular branch in the domain $1 < \left|z\right| < \infty$.

Determine, for what relationship between the numbers $\alpha_{1}, \alpha_{2}$ and $\alpha_{3}$, the multivalued function

admits the extraction of a regular branch in the domain $1 < \left|z\right| < \infty$.

Prove that in the plane with a cut along the negative part of the real axis there exists a regular branch of the multivalued function $\operatorname {Ln}(1-\sqrt{z})$, satisfying the condition $\varphi (2 i) = -\pi i / 2$.

Verify that the following multivalued functions admit the extraction of regular branches in the annulus $G$, and expand all the regular branches in a Laurent series in powers of $z-a$ in the annulus $G$ (the point $a$ and the annulus $G$ are indicated in parentheses):

Let $f(z)$ be a regular branch of the multivalued function $\operatorname {Ln}\left(\frac{z-1}{z+1}\right)$ in the domain $G = \left\{ z:\left|z\right| > 1\right\}$ such that $f(\infty ) = 0$. Prove that the multivalued function $\left\{ \sqrt{f(z)+4 i}\right\}$ splits in $G$ into regular branches.

Let $g(z)$ be a regular branch of the function $\left\{ \sqrt{f(z)+4 i}\right\}$ in $G$ such that $g(\infty ) = -\sqrt{2}(i+1)$. Compute the integral $\oint_{\left|z\right| = 2} \frac{d z}{g(z)}$.

Let $f(z)$ be a regular branch of the multivalued function $\left\{ \sqrt{\frac{3-z}{3+z}}\right\}$ in the domain $G = \left\{ z:\left|z\right| > 3\right\}$ such that $f(\infty ) = -i$. Prove that the multivalued function $\operatorname {Ln}(f(z)-i)$ splits in $G$ into regular branches.

Let $g(z)$ be a regular branch of the function $\operatorname {Ln}(f(z)-i)$ in $G$ such that $g(\infty ) = \ln 2-i \frac{\pi }{2}$. Compute the integral $\oint_{\left|z\right| = 4} \frac{d z}{g(z)}$.